Points A and B represent pressure sensors in fixed positions on the
base of a round tank. The chord through CD represents the water level
in the tank. Lines a and b are the heights of water registered by each
sensor...

You can draw a line of minimum distance between and perpendicular to two
lines in 3space. I know how to get the distance and direction of this
line, but I want to locate the line in 3space so that I can find its
midpoint.

Given two points on the earth's surface - their grid coordinates, and
their lat-long coordinates, and knowing how to get from the grid
coordinates to a grid bearing - how can I calculate the true bearing
between the two points?

I am looking to do a form of 'dead reckoning' using a fixed
latitude/longitude position, velocity components for north and west,
and a time delay to compute an extrapolated latitude/longitude position.

A right cylinder of radius 'r' (a lift duct) intercepts a plane (the
deck of a hovercraft) at an angle of 30 degrees to the vertical. How
may the resulting ellipse be scribed? - i.e., how do I mark the hole
to be cut in the deck?

What is the 'official' definition of 'edge'? Specifically, is an edge
restricted to the intersection of two non-coplanar faces or do two-
dimensional shapes have edges? I'm also curious about a definition of
'face'. How many faces does a two-dimensional shape have?

I know the area of a sphere is 4phi(r^2), but I'm wondering how to
derive that formula. I know it should be done in cylindrical
coordinates, and I'm thinking that the arc of a circle is defined as
rd(theta) and it's multiplied with rd(phi) to get (r^2)d(theta)d(phi).
Could you please help explain this?

If an ellipsoid has half-axes a, b, and c, and the plane is normal to the
vector [i,j,k] and also passes through the point (i,j,k), what are the
half-axes and orientation of the ellipse of intersection?

Given two points in 3-D space, such as A(x1,y1,z1) and B(x2,y2,z2),
what would be the equation of the line that connects those points? I
know that in the 2-D plane the equation of a line in slope-intercept
form is y = mx + b. Is there something similar in 3-D?

I am looking for the equation of an ellipse in 3-dimensional space. It
can be a parametric formulation (e.g., x(t), y(t), z(t)) or a more
canonical form (e.g., the 3D analog to the 2D form ((X*X)/a)+((Y*Y)/
b)=1).

I have a height map of a terrain where the x and y values are fixed. I
can calculate best-fit slopes in the x and y directions, but I can't
figure out how to combine them into a best-fit regression plane.

I have solar collectors on my roof. They are mounted so that the base
of each panel runs up the slope of the roof, and the panels themselves
are mounted at an angle. I'd like to know how to determine the various
angles created by this situation.